Riesz projector

In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum). It was introduced by Frigyes Riesz in 1912.

Definition

Let $A$ be a closed linear operator in the Banach space $𝔅$. Let $\Gamma$ be a simple or composite rectifiable contour, which encloses some region $G_{\Gamma }$ and lies entirely within the resolvent set $\rho (A)$ ($\Gamma \subset \rho (A)$) of the operator $A$. Assuming that the contour $\Gamma$ has a positive orientation with respect to the region $G_{\Gamma }$, the Riesz projector corresponding to $\Gamma$ is defined by

$P_{\Gamma }=-\frac{1}{2\pi \mathrm{i}}{\oint }_{\Gamma }(A-z{I}_{𝔅}{)}^{-1}\mathrm{d}z;$

here $I_{𝔅}$ is the identity operator in $𝔅$.

If $\lambda \in \sigma (A)$ is the only point of the spectrum of $A$ in $G_{\Gamma }$, then $P_{\Gamma }$ is denoted by $P_{\lambda }$.

Properties

The operator $P_{\Gamma }$ is a projector which commutes with $A$, and hence in the decomposition

$𝔅={𝔏}_{\Gamma }\oplus {𝔑}_{\Gamma }{𝔏}_{\Gamma }=P_{\Gamma }𝔅,{𝔑}_{\Gamma }=(I_{𝔅}-P_{\Gamma })𝔅,$

both terms ${𝔏}_{\Gamma }$ and ${𝔑}_{\Gamma }$ are invariant subspaces of the operator $A$. Moreover,

1. The spectrum of the restriction of $A$ to the subspace ${𝔏}_{\Gamma }$ is contained in the region $G_{\Gamma }$;
2. The spectrum of the restriction of $A$ to the subspace ${𝔑}_{\Gamma }$ lies outside the closure of $G_{\Gamma }$.

If ${\Gamma }_1$ and ${\Gamma }_2$ are two different contours having the properties indicated above, and the regions $G_{{\Gamma }_1}$ and $G_{{\Gamma }_2}$ have no points in common, then the projectors corresponding to them are mutually orthogonal:

$P_{{\Gamma }_1}{P}_{{\Gamma }_2}=P_{{\Gamma }_2}{P}_{{\Gamma }_1}=0.$

See also

• Spectrum (functional analysis)
• Decomposition of spectrum (functional analysis)
• Spectrum of an operator
• Resolvent formalism
• Operator theory